Loglinear Models for Categorical Data - YorkU Math … Models for Categorical Data outline Lecture...

Loglinear Models for Categorical Data

Sex: Male

Adm

it?: Y

es

Sex: Female

Adm

it?: N

o

1198 1493

557 1278

High

2

3

Low

High 2 3 Low

Rig

ht

Ey

e G

ra

de

Left Eye Grade

Unaided distant vision data

4.4

-3.1

2.3

-5.9

-2.2

7.0

Black Brown Red Blond

Bro

wn

Ha

ze

l G

ree

n

Blu

e

Michael FriendlyYork University, <[email protected]>

SPIDA 2004

Loglinear Models for Categorical Data outline

Lecture Outline

Introduction

Graphical methods for Categorical Data

Loglinear models: Prelimaries

Overview of loglinear models

Visualizing contingency tables and loglinear models

2 × 2 tables

Larger tables

n-way tables

Structured tables

Ordinal variables

Square tables

SPIDA 2004 1 Michael Friendly

Loglinear Models for Categorical Data outline

Resources

SAS macro programs, from SAS System for Statistical Graphics (Friendly, 1991):

http://www.math.yorku.ca/SCS/sssg/

http://www.math.yorku.ca/SCS/sasmac/

SAS macro programs, from Visualizing Categorical Data (Friendly, 2000):

http://www.math.yorku.ca/SCS/vcd/

Workshop notes: http://www.math.yorku.ca/SCS/spida/loglin/

SCS Short Course notes, Categorical Data Analysis with Graphics

http://www.math.yorku.ca/SCS/Courses/grcat/


Loglinear Models for Categorical Data overview2

Graphical Methods for Categorical Data

If I can’t picture it, I can’t understand it. Albert Einstein

Getting information from a table is like extracting sunlight from a cucumber.

Farquhar & Farquhar, 1891

Tables vs. Graphs

Tables are best suited for look-up— read off exact numbers

Graphs are better for showing patterns, trends, anomalies, making

comparisons

Visual presentation as communication: what do you want to say?



Visual metaphors

Quantitative data: magnitude ∼ position along an axis

Frequency data: count ∼ area (Friendly, 1995)

Sex: Male

Adm

it?: Y

es

Sex: Female

Adm

it?: N

o

1198 1493

557 1278

Fourfold display for 2×2 table

A

B

C

D

E

F

Male Female Admitted Rejected

Model: (DeptGender)(Admit)

Mosaic plot for 3-way table


LoglinearModelsforCategoricalDataoverview2

PrinciplesofGraphicalDisplays

Effectordering(FriendlyandKwan,2003)—Intablesandgraphs,sort

unorderedfactorsaccordingtotheeffectsyouwanttosee/show.

Auto data: Alpha order

Displa

Gratio

Hroom

Length

MPG

Price

Rep77

Rep78

Rseat

Trunk

Turn

Weight

Weig

htTurn

Tru

nk

Rseat

Rep78

Rep77

Pric

e M

PG

LengthH

room

Gra

tioDis

pla

Auto data: PC2/1 order

Gratio

MPG

Rep78

Rep77

Price

Hroom

Trunk

Rseat

Length

Weight

Displa

Turn

Turn

Dis

plaW

eig

ht

LengthR

seat

Tru

nk

Hro

om

Pric

e Rep77

Rep78

MP

G

Gra

tio

“Corrgrams:Exploratorydisplaysforcorrelationmatrices”(Friendly,2002)

SPIDA20045MichaelFriendly


Comparisons— Make visual comparisons easy

• Visual grouping— connect with lines, make key comparisons contiguous

• Baselines— compare data to model against a line, preferably horizontal

Fre

quen

cy

0

25

50

75

100

125

150

175

Number of Occurrences

0 1 2 3 4 5 6

Sqr

t(fr

eque

ncy)

-2

0

2

4

6

8

10

12

Number of Occurrences

0 1 2 3 4 5 6



Small multiples— combine stratified graphs into coherent displays (Tufte, 1983)

e.g., scatterplot matrix for quantitative data: all pairwise scatterplots

Prestige

14.8

87.2

Educ

6.38

15.97

Income

611

25879

Women

0

97.51



e.g., mosaic matrix for quantitative data: all pairwise mosaic plots

Admit

Male Female

Adm

it

Reje

ct

A B C D E F

Adm

it

Reje

ct

Admit Reject

Male

Fem

ale

Gender

A B C D E F

Male

Fem

ale

Admit Reject

A B

C

D

E

F

Male Female

A B

C

D

E

F

Dept


Loglinear Models for Categorical Data prelim

Loglinear models: Preliminaries

Categorical data:

Variables with a finite number of discrete values

• Binary: pass/fail, live/die, vote yes/vote no

• Nominal: unordered categories

• Religion: Christian, Jewish, Muslim, ...

• Party preference: Conservative, Liberal, NDP, Bloc Quebecois, Green

Party

• Ordinal: ordered categories

• Education: Primary, Secondary, University, Post-graduate

• Income ranges: 0-20, 20-30, 30-40, ...

• Likert scale: strongly disagree · · · strongly agree

• Interval:

• Counts: number of children– 0, 1, 2, ...




Data representations:

Case form: One row per case (observation), one column per variable—

standard data table

(Quantitative data is always represented in case form)

Table form: A set of n variables represented as an n-way table

Table 1: Admissions to Berkeley graduate programs

Admitted Rejected Total

Males 1198 1493 2691

Females 557 1278 1835

Total 1755 2771 4526



3+-way tables

Table 2: Berkeley data: 3-way table

Male Female

Dept Admit Reject Admit Reject

A 512 313 89 19

B 353 207 17 8

C 120 205 202 391

D 138 279 131 244

E 53 138 94 299

F 22 351 24 317



Frequency form: One row per combination of variable categories

1 Obs Admit Gender Dept count2

3 1 Admitted Male A 5124 2 Admitted Male B 3535 3 Admitted Male C 1206 4 Admitted Male D 1387 5 Admitted Male E 538 6 Admitted Male F 229 7 Admitted Female A 89

10 8 Admitted Female B 1711 9 Admitted Female C 20212 10 Admitted Female D 13113 11 Admitted Female E 9414 12 Admitted Female F 2415 ... ... ... ... ...16 23 Rejected Female E 29917 24 Rejected Female F 317




Data transformations:

All SAS procedures take input in case form.

Most SAS procedures take input in frequency form— need to use a FREQ or

WEIGHT statement to identify the frequency (count) variable.

Categorical variables can be represented by numeric or character values

• Character variables are ordered alphabetically by default— often not what

you want (e.g., Low, Medium, High → High, Low, Medium)

• Often easiest to use numeric values, and create a SAS format to attach value

labels:1 proc format;2 value inc 1=’Low’ 2=’Medium’ 3=’High’;3 proc freq;4 tables educ * income;5 format income inc.;



Recoding categories

• Categories can be collapsed (frequencies summed) using SAS formats1 proc format;2 value edu 1-8=’Primary’ 9-12=’Secondary’3 12-16=’University’ 16-high=’Post-grad’;4 proc freq;5 tables educ * income;6 format educ edu. income inc.;

Reordering categories

• Many SAS procedures allow several different ways to order the categories of

a discrete variable

• ORDER=INTERNAL— by numeric or character value (default).

• ORDER=DATA— by order of appearance in the dataset.

• ORDER=FORMATTED— by order of formatted value.

• ORDER=FREQ— by order of frequency of occurrence.

•The table macro

• Collapse or recode variables


Loglinear Models for Categorical Data overview

Loglinear models: Overview

Related perspectivesLoglinear models can be developed as an analog of classical ANOVA andregression models, where multiplicative relations (under independence) arere-expressed in additive form as models for log(frequency).Alternatively, but somewhat more generally, loglinear models are also GLMs forlog(frequency), with a Poisson distribution for the cell counts.

Two-way tables: Loglinear approachFor two discrete variables, A and B, suppose a multinomial sample of totalsize n over the IJ cells of a two-way I × J contingency table, with cellfrequencies nij , and cell probabilities πij = nij/n.• The table variables are statistically independent when the cell (joint)

probability equals the product of the marginal probabilities,Pr(A = i & B = j) = Pr(A = i) × Pr(B = j), or,

πij = πi+π+j .

• An equivalent model in terms of expected frequencies, mij = nπij is

mij = n mi+ m+j .



• This multiplicative model can be expressed in additive form as a model for

log mij ,

log mij = log n + log mi+ + log m+j . (1)

• By anology with ANOVA models, the independence model (1) can be

expressed as

log mij = µ + λAi + λB

j , (2)

where µ is the grand mean of log mij and the parameters λAi and λB

j

express the marginal frequencies of variables A and B, and are typically

defined so that∑

i λAi =

∑j λB

j = 0.

Dependence between the table variables is expressed by adding association

parameters, λABij , giving the saturated model,


j + λABij . (3)

• The saturated model fits the table perfectly (mij = nij ): there are as many

parameters as cell frequencies.

• A global test for association tests H0 : λABij = 0.

• For ordinal variables, the λABij may be structured more simply, giving tests

for ordinal association.



Two-way tables: GLM approach

In the GLM approach, the vector of cell frequencies, n = {nij} is specified to

have a Poisson distribution with means m = {mij} given by

log m = Xβ

where X is a known design (model) matrix and β is a column vector

containing the unknown λ parameters.

For example, for a 2 × 2 table, the saturated model (3) with the usual zero-sum

constraints can be represented as

log m11

log m12

log m21

log m22

=

1 1 1 1

1 1 −1 −1

1 −1 1 −1

1 −1 −1 1

µ

λA1

λB1

λAB11

Note that only the linearly independent parameters are represented.

λA2 = −λA

1 , because λA1 + λA

2 = 0, and so forth.

An advantage of the GLM formulation is that it makes it somewhat easier to

express models with ordinal or quantitative variables.



Three-way Tables

Saturated model: For a 3-way table, of size I × J × K for variables

A, B, C , the saturated loglinear model includes associations between all pairs of

variables, as well as a 3-way association term, λABCijk

log mijk = µ + λAi + λB

j + λCk + λAB

ij + λACik + λBC

jk + λABCijk . (4)

As before, one-way terms (λAi , λB

j , λCk ) pertain to differences in the marginal

frequencies of the table variables.

Two-way terms (λABij , λAC

ik , λBCjk ) pertain to the partial association for each

pair of variables, controlling for the remaining variable.

The three-way term, λABCijk allows the partial association between any pair of

variables to vary over the categories of the third variable.

Such models are usually hierarchical : the presence of a high-order term, such

as λABCijk implies that all low-order relatives are automatically included.

Thus, a short-hand notation for a loglinear model lists only the high-order

terms, i.e., model (4) ≡ [ABC]



Reduced models: The usual goal is to fit the smallest model (fewest

high-order terms) that is sufficient to explain/describe the observed frequencies.

All representative subset models are shown in the table below

Table 3: Log-linear Models for Three-Way Tables

Model Model symbol Interpretation

Mutual independence [A][B][C] A ⊥ B ⊥ C

Joint independence [AB][C] (A B) ⊥ C

Conditional independence [AC][BC] (A ⊥ B) |CAll two-way associations [AB][AC][BC] homogeneous association

Saturated model [ABC] interaction

e.g.,

[AB][C] ≡ log mijk = µ + λAi + λB

j + λCk + λAB

ij



Assessing goodness of fit

Goodness of fit of a specified model may be tested by the likelihood ratio G2,

G2 = 2∑

i

ni log(ni/mi) , (5)

or the Pearson χ2,

χ2 =∑

i

(ni − mi)2

mi, (6)

with degrees of freedom = # cells - # estimated parameters.

E.g., for the model of mutual independence, [A][B][C], df =

IJK − (I − 1) − (J − 1) − (K − 1) = (I − 1)(J − 1)(K − 1)The terms summed in (5) and (6) are the squared cell residuals

Other measures of balance goodness of fit against parsimony, e.g., Akaike’s

Information Criterion (smaller is better)

AIC = G2 − 2df or AIC = G2 + 2 # parameters


Loglinear Models for Categorical Data vismethods

Visualizing Contingency tables

Two-way tables

2 × 2 tables — Visualize odds ratio (FFOLD macro)2 × 2 × k tables — Homogeneity of associationr × 3 tables — Trilinear plots (TRIPLOT macro)r × c tables — Visualize association (SIEVE program)r × c tables — Visualize association (MOSAIC macro)Square r × r tables — Visualize agreement (AGREE program)

n-way tables

Fit loglinear models, visualize lack-of-fit — (MOSAIC macro)Test & visualize partial association — (MOSAIC macro)Visualize pairwise association — (MOSMAT macro)Visualize conditional association — (MOSMAT macro)Visualize loglinear structure — (MOSMAT macro)

Correspondence analysis and MCA — (CORRESP macro)


Loglinear Models for Categorical Data 2by2

2 × 2 tables

2 × 2 tables provide the simplest context to illustrate loglinear models and theirrelations to other tests and modeling techniques.

Example: Bickel et al. (1975): admissions to graduate depatments at U. C.Berkeley in 1973— Aggregate data for the six largest departments:

Table 4: Admissions to Berkeley graduate programs

Admitted Rejected Total % Admitted

Males 1198 1493 2691 44.52

Females 557 1278 1835 30.35

Total 1755 2771 4526 38.78

Equivalent questions:

Proportions: Are the proportions of males and females admitted the same?

• Method: Test of independent proportions, H0 : p1 = p2

• Answer:z = p1−p2√

p(1−p)(1/n1+1/n2)= .4452−.3035√

.3878(.6163)(1/2691+1/1835)= 9.62



2 × 2 tables

Odds: Are the odds of admission the same for males and females?

• Method: Odds ratio, H0 : θ = Odds(Admit |Male)

Odds(Admit |Female)= 1

• Answer: θ = Odds(Admit |Male)

Odds(Admit |Female)= 1198/1493

557/1276 = 1.84

• → Males 84% more likely to be admitted.

Is admission independent of gender?

• Method: χ2 test of independence, H0 : pij = pi·p·j

• Answer: χ2(1) =

∑ ∑ (nij−mij)2

mij= 92.21, p < 0.0001

Does cell frequency depend only on marginal frequencies of gender and

admission?

• Method: Loglinear model, log(mij) = µ + λAi + λG

j + λAGij

• Answer: H0 : λAGij = 0 LR G2

(1) = 93.45, p < 0.0001



Standard analysis: PROC FREQ

proc freq data=berkeley;weight freq;tables gender*admit / chisq;

Output:

Statistics for Table of gender by admit

Statistic DF Value Prob------------------------------------------------------Chi-Square 1 92.2053 <.0001Likelihood Ratio Chi-Square 1 93.4494 <.0001Continuity Adj. Chi-Square 1 91.6096 <.0001Mantel-Haenszel Chi-Square 1 92.1849 <.0001Phi Coefficient 0.1427

How to visualize and interpret?



Fourfold displays for 2 × 2 tables

Quarter circles: radius ∼ √nij ⇒ area ∼ frequency

Independence: Adjoining quadrants ≈ alignOdds ratio: ratio of areas of diagonally opposite cellsConfidence rings: Visual test of H0 : θ = 1 ↔ adjoining rings overlap

Sex: Male

Adm

it?: Y

es

Sex: Female

Adm

it?: N

o

1198 1493

557 1278

Confidence rings do not overlap: θ 6= 1



Fourfold displays for 2 × 2 × k tables

Data in Table 4 had been pooled over departmentsStratified analysis: one fourfold display for each departmentEach 2 × 2 table standardized to equate marginal frequenciesShading: highlight departments for which Ha : θi 6= 1

Sex: Male

Ad

mit?

: Y

es

Sex: Female

Ad

mit?

: N

o

512 313

89 19

Department: A Sex: Male

Ad

mit?

: Y

es

Sex: Female

Ad

mit?

: N

o

353 207

17 8

Department: B Sex: Male

Ad

mit?

: Y

es

Sex: Female

Ad

mit?

: N

o

120 205

202 391

Department: C

Sex: Male

Ad

mit?

: Y

es

Sex: Female

Ad

mit?

: N

o

138 279

131 244

Department: D Sex: Male

Ad

mit?

: Y

es

Sex: Female

Ad

mit?

: N

o

53 138

94 299

Department: E Sex: Male

Ad

mit?

: Y

es

Sex: Female

Ad

mit?

: N

o

22 351

24 317

Department: F

Only one department (A) shows association; θA = 0.349 → women(0.349)−1 = 2.86 times as likely as men to be admitted.



What happened here?

Simpson’s paradox:

Aggregate data are misleading because they falsely assume men and women

apply equally in each field.

But:

Large differences in admission rates across departments.

Men and women apply to these departments differentially.

Women applied in large numbers to departments with low admission rates.

(This ignores possibility of structural bias against women: differential funding of

fields to which women are more likely to apply.)

Other graphical methods can show these effects.



The FOURFOLD program and the FFOLD macro

The FOURFOLD program is written in SAS/IML.The FFOLD macro provides a simpler interface.Printed output: (a) significance tests for individual odds ratios, (b) tests ofhomogeneity of association (here, over departments) and (c) conditionalassociation (controlling for department).

Plot by department:berk4f.sas

1 %include catdata(berkeley);2

3 %ffold(data=berkeley,4 var=Admit Gender, /* panel variables */5 by=Dept, /* stratify by dept */6 down=2, across=3, /* panel arrangement */7 htext=2); /* font size */

Aggregate data: first sum over departments, using the TABLE macro:

8 %table(data=berkeley, out=berk2,9 var=Admit Gender, /* omit dept */

10 weight=count, /* frequency variable */11 order=data);12 %ffold(data=berk2, var=Admit Gender);



The FOURFOLD program and the FFOLD macro

Printed output:

Odds (Admitted|Male) / (Admitted|Female)

Odds Ratio Log Odds SE(LogOdds) Z Pr>|Z|

A 0.349 -1.052 0.263 -4.005 0.000B 0.803 -0.220 0.438 -0.503 0.615C 1.133 0.125 0.144 0.868 0.385D 0.921 -0.082 0.150 -0.546 0.585E 1.222 0.200 0.200 1.000 0.317F 0.828 -0.189 0.305 -0.619 0.536



For a 2 × 2 × k table, additional tests are also of interest:

Homogeneity of odds ratios: H0 : θ1 = θ2 = · · · θk (This is equivalent to theloglinear model of no 3-way association, [AG][AD][DG].)

Conditional independence: Admit ⊥ Gender |Dept

Test of Homogeneity of Odds Ratios (no 3-Way Association)TEST CHISQ DF PROB

Homogeneity of Odds Ratios 20.204 5 0.0011

Conditional Independence of Admit and Gender | Dept(assuming Homogeneity)

TEST CHISQ DF PROB

Likelihood-Ratio 1.531 1 0.2159Cochran-Mantel-Haenszel 1.429 1 0.2320


Loglinear Models for Categorical Data twoway

Two-way frequency tables

Table 5: Hair-color eye-color data

Eye Hair Color

Color Black Brown Red Blond Total

Green 5 29 14 16 64

Hazel 15 54 14 10 93

Blue 20 84 17 94 215

Brown 68 119 26 7 220

Total 108 286 71 127 592



Two-way frequency tables: PROC FREQ

proc freq data=haireye;weight count;tables Eye * HAIR / chisq;

Output:

Statistics for Table of Eye by HAIR

Statistic DF Value Prob------------------------------------------------------Chi-Square 9 138.2898 <.0001Likelihood Ratio Chi-Square 9 146.4436 <.0001Mantel-Haenszel Chi-Square 1 28.2923 <.0001Phi Coefficient 0.4833



Two-way frequency tables: PROC GENMOD

We can fit the same model as a Poisson GLM for count:

proc genmod data=haireye;class hair eye;model count = hair eye hair*eye / dist=poisson type3;

Output:

LR Statistics For Type 3 Analysis

Chi-Source DF Square Pr > ChiSq

hair 3 143.46 <.0001eye 3 58.57 <.0001hair*eye 9 146.44 <.0001

How to visualize and interpret?



Two-way frequency tables: Sieve diagrams

count ∼ areaWhen row/col variables are independent, nij ∼ ni+n+j

⇒ each cell can be represented as a rectangle, with area = height × width ∼frequency, nij

Green

Hazel

Blue

Brown


Eye

Co

lor

Hair Color

Expected frequencies: Hair Eye Color Data11.7 30.9 7.7 13.7

17.0 44.9 11.2 20.0

39.2 103.9 25.8 46.1

40.1 106.3 26.4 47.2

64

93

215

220

108 286 71 127 592



Sieve diagrams

Height/width ∼ marginal frequencies, ni+, n+j

Area ∼ expected frequency, ∼ ni+n+j

Shading ∼ observed frequency, nij , color: sign(nij − mij).Independence: Shown when density of shading is uniform.

Green

Hazel

Blue

Brown


Eye

Co

lor

Hair Color

5 29 14 16

15 54 14 10

20 84 17 94

68 119 26 7



Sieve diagrams

Effect ordering: Reorder rows/cols to make the pattern coherent

Blue

Green

Hazel

Brown


Eye

Co

lor

Hair Color

20 84 17 94

5 29 14 16

15 54 14 10

68 119 26 7



Sieve diagrams

Vision classification data for 7477 women

High

2

3

Low

High 2 3 Low

Rig

ht

Eye G

rad

e

Left Eye Grade

Unaided distant vision data


Loglinear Models for Categorical Data nway2

Mosaic displays and Log-linear Models

Hartigan and Kleiner (1981), Friendly (1994, 1999):

Width ∼ one set of marginals, ni+

Height ∼ relative proportions of other variable, pj | i = nij/ni+

⇒ area ∼ frequency, nij = ni+pj | i

1198

1493

557

1278

Male Female

Adm

itte

dR

eje

cte

dModel: (Gender)(Admit)

Sta

ndard

ized

resid

uals

:

<-4

-4:-

2-2

:-0

0:2

2:4

>4



Shading: Sign and magnitude of Pearson χ2 residual,dij = (nij − mij)/

√mij (or L.R. G2)

Sign: − negative in red; + positive in blueMagnitude: intensity of shading: |dij | > 0, 2, 4, . . .

Independence: Rows ≈ align, or cells are empty!

E.g., aggregate Berkeley data, independence model:

1198

1493

557

1278

Male Female

Adm

itte

dR

eje

cte

dModel: (Gender)(Admit)

Sta

ndard

ized

resid

uals

:

<-4

-4:-

2-2

:-0

0:2

2:4

>4



Mosaic displays

Departments × Gender:

Did departments differ in the total number of applicants?

Did men and women apply differentially to departments?

A

B

C

D

E

F

Male Female

Model: (Dept)(Gender)

Model [Dept] [Gender]: G2(5) =

1220.6.Note: Departments ordered A–F byoverall rate of admission.



Mosaic displays: Hair color and eye color

4.4

-3.1

2.3

-5.9

-2.2

7.0


Bro

wn

Hazel G

reen B

lue

Dark hair goes with dark eyes, light

hair with light eyes

Red hair, hazel eyes an exception?

Effect ordering: Rows/cols permuted

by CA Dimension 1



Mosaic displays for multiway tables

Generalizes to n-way tables: divide cells recursively

Can fit any log-linear model (e.g., 3-way),

Table 6: Log-linear Models for Three-Way Tables

Model Model symbol Independence interpretation

Mutual independence [A][B][C] A ⊥ B ⊥ C

Joint independence [AB][C] (A B) ⊥ C

Conditional independence [AC][BC] (A ⊥ B) |CAll two-way associations [AB][AC][BC] (none)

Saturated model [ABC] (none)

e.g., the model for conditional independence (A ⊥ C |B):

[AB][BC] ≡ log mijk = µ + λAi + λB

j + λCk + λAB

ij + λBCjk

Each mosaics shows:

DATA (size of tiles)(some) marginal frequencies (spacing → visual grouping)RESIDUALS (shading) — what associations have been omitted?



E.g., Joint independence, [DG][A] (null model, Admit as response) [G2(11) = 877.1]:

A

B

C

D

E

F





Mosaic displays for multiway tables

Visual fitting:

Pattern of lack-of-fit (residuals) → “better” model— smaller residuals“cleaning the mosaic” → “better” model— empty cellsbest done interactively!

-4.2 4.2 4.2 -4.2A

B

C

D

E

F


Model: (DeptGender)(DeptAdmit)

E.g., Add [Dept Admit] association →Conditional independence:

Fits poorly, overall (G2(6) = 21.74)

But, only in Department A!



Sequential plots and models

Mosaic for an n-way table → hierarchical decomposition of association in a way

analogous to sequential fitting in regression

Joint cell probabilities are decomposed as

pijk`··· =

{v1v2}︷︸︸︷pi × pj|i × pk|ij︸︷︷︸

{v1v2v3}

× p`|ijk × · · · × pn|ijk···

First 2 terms → mosaic for v1 and v2

First 3 terms → mosaic for v1, v2 and v3

· · ·Sequential models of joint independence → additive decomposition of the total

association, G2[v1][v2]...[vp] (mutual independence),

G2[v1][v2]...[vp] = G2

[v1][v2]+G2

[v1v2][v3]+G2

[v1v2v3][v4]+ · · ·+G2

[v1...vp−1][vp]



Sequential plots and models: Example

Hair color x Eye color marginal table (ignoring Sex)


Bro

wn

H

azel

Gre

en

Blu

e

(Hair)(Eye), G2 (9) = 146.44




3-way table, Joint Independence Model [Hair Eye] [Sex]


Bro

wn

H

azel

Gre

en

Blu

e

M F

(HairEye)(Sex), G2 (15) = 19.86




3-way table, Mutual Independence Model [Hair] [Eye] [Sex]


Bro

wn

H

azel

Gre

en

Blu

e

M F

(Hair)(Eye)(Sex), G2 (24) = 166.30





Bro

wn

H

azel

Gre

en

Blu

e

(Hair)(Eye), G2 (9) = 146.44

[Hair] [Eye]

G2(9) = 146.44

+


Bro

wn

H

azel

Gre

en

Blu

e

M F

(HairEye)(Sex), G2 (15) = 19.86

[Hair Eye] [Sex]

G2(15) = 19.86

=


Bro

wn

H

azel

Gre

en

Blu

e

M F

(Hair)(Eye)(Sex), G2 (24) = 166.30

[Hair] [Eye] [Sex]

G2(24) = 166.60



Mosaic matrices

Analog of scatterplot matrix for categorical data (Friendly, 1999)

Shows all p(p − 1) pairwise views in a coherent display

Each pairwise mosaic shows bivariate (marginal) relation

Fit: marginal independence

Residuals: show marginal associations

Direct visualization of the “Burt” matrix analyzed in multiple correspondence

analysis for p categorical variables

Hair

Brown Haz Grn Blue

Bla

ck

Bro

wn R

ed Blo

nd

Male Female

Bla

ck

Bro

wn

Red Blo

nd


Bro

wn

Haz

Grn

Blu

e

Eye

Male Female

Bro

wn

Haz

Grn

B

lue


Male

F

em

ale

Brown Haz Grn Blue

Male

F

em

ale

Sex



Hair

Brown Haz Grn Blue

Bla

ck

Bro

wn R

ed Blo

nd

Male Female

Bla

ck

Bro

wn

Red Blo

nd


Bro

wn

Haz

Grn

Blu

e

Eye

Male Female

Bro

wn

Haz

Grn

B

lue


Male

F

em

ale

Brown Haz Grn Blue

Male

F

em

ale

Sex



Berkeley data:

Admit

Male Female

Adm

it

Rej

ect

A B C D E F

Adm

it

Rej

ect

Admit Reject

Mal

e

Fem

ale

Gender

A B C D E F

Mal

e

Fem

ale

Admit Reject

A

B

C

D

E

F

Male Female

A

B

C

D

E

F

Dept



Partial association, Partial mosaics

Stratified analysis:How does the association between two (or more) variables vary over levels ofother variables?Mosaic plots for the main variables show partial association at each level of theother variables.E.g., Hair color, Eye color BY Sex ↔ TABLES sex * hair * eye;

2.8

-2.1

-3.3

3.3


Bro

wn

Ha

zel

Gre

en

B

lue

Sex: Male

3.5

-2.3 -2.5

-4.9

-2.0

6.4


Bro

wn

Ha

zel

Gre

en

Blu

e

Sex: Female



Partial association, Partial mosaics

Stratified analysis:For models of partial independence, A ⊥ B at each level of (controlling for) CA ⊥ B |Ck, partial G2s add to the overall G2 for conditional independence,

G2A⊥B |C =

∑

k

G2A⊥B |C(k)

Table 7: Partial and Overall conditional tests, Hair ⊥ Eye |Sex

Model df G2 p-value

[Hair][Eye] | Male 9 44.445 0.000

[Hair][Eye] | Female 9 112.233 0.000

[Hair][Eye] | Sex 18 156.668 0.000



Software for Mosaic Displays

Demonstration web applet:http://www.math.yorku.ca/SCS/Online/mosaics/

Runs the current version of mosaics via a cgi-scriptCan run sample data, upload a data file, enter data in a form.Choose model fitting and display options (not all supported).


Loglinear Models for Categorical Data mossoft




SAS software & documentation:http://www.math.yorku.ca/SCS/mosaics.htmlhttp://www.math.yorku.ca/SCS/vcd/

Examples: Many in VCD and on web site

SAS/IML modules: mosaics.sas SAS/IML program

Enter frequency table directly in SAS/IML, or read from a SAS dataset.Most flexible:• Select, collapse, reorder, re-label table levels using SAS/IML statements• Specify structural 0s, fit specialized models (e.g., quasi-independence)• Interface to models fit using PROC GENMOD




Macro interface: mosaic macro, table macro, mosmat macro

mosaic macroEasiest to use:• Direct input from a SAS dataset• No knowledge of SAS/IML required• Reorder table variables; collapse, reorder table levels with table macro• Convenient interface to partial mosaics (BY=)

table macroCreate frequency table from raw dataCollapse, reorder table categoriesRe-code table categories using SAS formats, e.g., 1=’Male’ 2=’Female’

mosmat macroMosaic matrices— analog of scatterplot matrix (Friendly, 1999)



mosaic macro example: Berkeley data

berkeley.sas1 title ’Berkeley Admissions data’;2 proc format;3 value admit 1="Admitted" 0="Rejected" ;4 value dept 1="A" 2="B" 3="C" 4="D" 5="E" 6="F";5 value $sex ’M’=’Male’ ’F’=’Female’;6 data berkeley;7 do dept = 1 to 6;8 do gender = ’M’, ’F’;9 do admit = 1, 0;

10 input freq @@;11 output;12 end; end; end;13 /* -- Male -- - Female- */14 /* Admit Rej Admit Rej */15 datalines;16 512 313 89 19 /* Dept A */17 353 207 17 8 /* B */18 120 205 202 391 /* C */19 138 279 131 244 /* D */20 53 138 94 299 /* E */21 22 351 24 317 /* F */22 ;



Data set berkeley:

dept gender admit freq

1 M 1 5121 M 0 3131 F 1 891 F 0 192 M 1 3532 M 0 2072 F 1 172 F 0 83 M 1 1203 M 0 2053 F 1 2023 F 0 3914 M 1 1384 M 0 2794 F 1 1314 F 0 2445 M 1 535 M 0 1385 F 1 945 F 0 2996 M 1 226 M 0 3516 F 1 246 F 0 317




mosaic9m.sas1 goptions hsize=7in vsize=7in;2

3 %include catdata(berkeley);4

5 *-- apply character formats to numeric table variables;6 %table(data=berkeley,7 var=Admit Gender Dept,8 weight=freq,9 char=Y, format=admit admit. gender $sex. dept dept.,

10 order=data, out=berkeley);11

12 %mosaic(data=berkeley,13 vorder=Dept Gender Admit, /* reorder variables */14 plots=2:3, /* which plots? */15 fittype=joint, /* fit joint indep. */16 split=H V V, htext=3); /* options */




A

B

C

D

E

F

Male Female

Model: (Dept)(Gender)

Two-way, Dept. by Gender

A

B

C

D

E

F



Three-way, Dept. by Gender by Admit



mosmat macro: Mosaic matrices

mosmat9m.sas1 %include catdata(berkeley);2 %mosmat(data=berkeley,3 vorder=Admit Gender Dept, sort=no);

Admit

Male Female

Adm

it

Reje

ct

A B C D E F

Adm

it

Reje

ct

Admit Reject

Male

Fem

ale

Gender

A B C D E F

Male

Fem

ale

Admit Reject

A B

C

D

E

F

Male Female

A B

C

D

E

F

Dept



Partial mosaics

mospart3.sas1 %include catdata(hairdat3s);2 %gdispla(OFF);3 %mosaic(data=haireye,4 vorder=Hair Eye Sex, by=Sex,5 htext=2, cellfill=dev);6 %gdispla(ON);7 %panels(rows=1, cols=2); /* make 2 figs -> 1 */

2.8

-2.1

-3.3

3.3


Bro

wn

Ha

zel

Gre

en

B

lue

Sex: Male

3.5

-2.3 -2.5

-4.9

-2.0

6.4


Bro

wn

Ha

zel

Gre

en

Blu

e

Sex: Female


Loglinear Models for Categorical Data corresp

Correspondence analysis and MCA

Correspondence analysis (CA): Analog of PCA for frequency data:

account for maximum % of χ2 in few (2-3) dimensions

finds scores for row (xim) and column (yjm) categories on these dimensions

uses Singular Value Decomposition of residuals from independence,

dij = (nij − mij)/√

mij

dij√n

=M∑

m=1

λm xim yjm

optimal scaling: each pair of scores, xim) and column (yjm, have highest

possible correlation (= λm).

plots of the row (xim) and column (yjm) scores show associations

MCA: Extends CA to n-way tables, but only uses bivariate associations (like

mosaic matrix)



Hair color, Eye color data:

Brown Blue

Hazel

Green

BLACK

BROWN

RED

BLOND

* Eye color * HAIR COLORD

im 2

(9.

5%)

-0.5

0.0

0.5

Dim 1 (89.4%)

-1.0 -0.5 0.0 0.5 1.0

Interpretation: row/column points “near” each other are positively associated

Dim 1: 89.4% of χ2 (dark ↔ light)

Dim 2: 9.5% of χ2 (RED/Green vs. others)



PROC CORRESP and the CORRESP macro

Two forms of input dataset:dataset in contingency table form – column variables are levels of one factor,observations (rows) are levels of the other.

Obs Eye BLACK BROWN RED BLOND

1 Brown 68 119 26 72 Blue 20 84 17 943 Hazel 15 54 14 104 Green 5 29 14 16

Raw category responses (case form), or cell frequencies (frequency form),classified by 2 or more factors (e.g., output from PROC FREQ)

Obs Eye HAIR Count

1 Brown BLACK 682 Brown BROWN 1193 Brown RED 264 Brown BLOND 7...

15 Green RED 1416 Green BLOND 16



PROC CORRESP and the CORRESP macro

PROC CORRESP

Handles 2-way CA, extensions to n-way tables, and MCA

Many options for scaling row/column coordinates and output statistics

OUTC= option → output dataset for plotting (PROC CORRESP doesn’t do plots

itself)

CORRESP macro

Uses PROC CORRESP for analysis

Produces labeled plots of the category points in either 2 or 3 dimensions

Many graphic options; can equate axes automatically

See: http://www.math.yorku.ca/vcd/corresp.html



Example: Hair and Eye Color

Input the data in contingency table form

corresp2a.sas · · ·

1 data haireye;2 input EYE $ BLACK BROWN RED BLOND ;3 datalines;4 Brown 68 119 26 75 Blue 20 84 17 946 Hazel 15 54 14 107 Green 5 29 14 168 ;




Using PROC CORRESP directly— labeled printer plot

proc corresp data=haireye outc=coord short;id eye; /* row variable */var black brown red blond; /* col variables */

proc plot data=coord vtoh=2; /* plot step */plot dim2 * dim1 = ’*’ $eye/ box haxis=by .1 vaxis=by .1; /* plot options */

Using the CORRESP macro— labeled high-res plot

%corresp (data=haireye,id=eye, /* row variable */var=black brown red blond, /* col variables */dimlab=Dim); /* options */




Printed output:

The Correspondence Analysis Procedure

Inertia and Chi-Square Decomposition

Singular Principal Chi-Values Inertias Squares Percents 18 36 54 72 90

----+----+----+----+----+---0.45692 0.20877 123.593 89.37% *************************0.14909 0.02223 13.158 9.51% ***0.05097 0.00260 1.538 1.11%

------- -------0.23360 138.29 (Degrees of Freedom = 9)

Row CoordinatesDim1 Dim2

Brown -.492158 -.088322Blue 0.547414 -.082954Hazel -.212597 0.167391Green 0.161753 0.339040

Column CoordinatesDim1 Dim2

BLACK -.504562 -.214820BROWN -.148253 0.032666RED -.129523 0.319642BLOND 0.835348 -.069579




Output dataset(selected variables):

Obs _TYPE_ EYE DIM1 DIM2

1 INERTIA . .2 OBS Brown -0.49216 -0.088323 OBS Blue 0.54741 -0.082954 OBS Hazel -0.21260 0.167395 OBS Green 0.16175 0.339046 VAR BLACK -0.50456 -0.214827 VAR BROWN -0.14825 0.032678 VAR RED -0.12952 0.319649 VAR BLOND 0.83535 -0.06958

Row and column points are distinguished by the _TYPE_ variable: OBS vs. VAR




Graphic output from CORRESP macro:

Brown Blue

Hazel

Green

BLACK

BROWN

RED

BLOND

* Eye color * HAIR COLOR

Dim

2 (

9.5%

)

-0.5

0.0

0.5

Dim 1 (89.4%)

-1.0 -0.5 0.0 0.5 1.0

Top legend produced with Annotate data set and the INANNO= option to theCORRESP macro



Multi-way tables

Stacking approach: van der Heijden and de Leeuw (1985)—

three-way table, of size I × J × K can be sliced and stacked as a two-waytable, of size (I × J) × K

K

J

I

I x J x K table

K

J

J

J

(I J )x K table

The variables combined are treated “interactively”Each way of stacking corresponds to a loglinear model• (I × J) × K → [AB][C]• I × (J × K) → [A][BC]• J × (I × K) → [B][AC]



Multi-way tables: Stacking

PROC CORRESP: Use TABLES statement and option CROSS=ROW orCROSS=COL. E.g., for model [A B] [C],

proc corresp cross=row;tables A B, C;weight count;

CORRESP macro: Can use / instead of ,

%corresp(options=cross=row,tables=A B/ C,weight count);



Example: Suicide Rates

Suicide rates in West Germany, by Age, Sex and Method of suicide

Sex Age POISON GAS HANG DROWN GUN JUMP

M 10-20 1160 335 1524 67 512 189M 25-35 2823 883 2751 213 852 366M 40-50 2465 625 3936 247 875 244M 55-65 1531 201 3581 207 477 273M 70-90 938 45 2948 212 229 268

F 10-20 921 40 212 30 25 131F 25-35 1672 113 575 139 64 276F 40-50 2224 91 1481 354 52 327F 55-65 2283 45 2014 679 29 388F 70-90 1548 29 1355 501 3 383

CA of the [Age Sex] by [Method] table:

Shows associations between the Age-Sex combinations and Method

Ignores association between Age and Sex



Example: Suicide Rates

suicide5.sas · · ·

1 %include catdata(suicide);2 *-- equate axes!;3 axis1 order=(-.7 to .7 by .7) length=6.5 in label=(a=90 r=0);4 axis2 order=(-.7 to .7 by .7) length=6.5 in;5 %corresp(data=suicide, weight=count,6 tables=%str(age sex, method),7 options=cross=row short,8 vaxis=axis1, haxis=axis2);

Output:

Inertia and Chi-Square Decomposition

Singular Principal Chi-Values Inertias Squares Percents 12 24 36 48 60

----+----+----+----+----+---0.32138 0.10328 5056.91 60.41% *************************0.23736 0.05634 2758.41 32.95% **************0.09378 0.00879 430.55 5.14% **0.04171 0.00174 85.17 1.02%0.02867 0.00082 40.24 0.48%

------- -------0.17098 8371.28 (Degrees of Freedom = 45)



10-20 * F

10-20 * M

25-35 * F25-35 * M

40-50 * F 40-50 * M

55-65 * F

55-65 * M

70-90 * F

70-90 * M

Drown

Gas

Gun

Hang

Jump

Poison

Dim

en

sio

n 2

(3

3.0

%)

-0.7

0.0

0.7

Dimension 1 (60.4%)-0.7 0.0 0.7



Compare with mosaic display:

MA

LE

F

EM

AL

E

10-20 25-35 40-50 55-65 >65

Gu

n

Ga

s

Ha

ng

P

ois

on

Ju

mp

Dro

wn

Suicide data - Model (SexAge)(Method)

Sta

nd

ard

ize

dre

sid

ua

ls:

<

-8-8

:-4

-4:-

2-2

:-0

0:2

2:4

4:8

>8


Loglinear Models for Categorical Data structured

More structured tables

Ordered categories

Tables with ordered categories allow more parsimonious representations of

association

Can represent λABij by a small number of parameters

→ more focused and more powerful tests of lack of independence

Square tables

For square I × I tables, where row and column variables have the same

categories:

• Can ignore diagonal cells, where association is expected and test remaining

association (quasi-independence)

• Can test whether association is symmetric around the diagonal cells.



Ordered categories

Ordinal scores

In many cases it may be reasonable to assign numeric scores, {ai} to an

ordinal row variable and/or numeric scores, {bi} to an ordinal column variable.

Typically, scores are equally spaced and sum to zero, {ai} = i − (I + 1)/2,

e.g., {ai} = {−1, 0, 1} for I=3.

Linear-by-Linear (Uniform) Association: When both variables are

ordinal, the simplest model posits that any association is linear in both variables.

λABij = γ aibj

This model adds one additional parameter to the independence model (γ = 0).

It is similar to

For integer scores, the local log odds ratio for any contiguous 2 × 2 table are

all equal, log θij = γThis is a model of uniform association



Row Effects and Column Effects: When only one variable is assigned

scores, we have the row effects model or the column effects model.

E.g., in the row effects model, the row variable (A) is treated as nominal, while

the column variable (B) is assigned ordered scores {bj}.


j + αibj

where the row parameters, αi, are defined so they sum to zero.

This model has (I − 1) more parameters than the independence model.

A Row Effects + Column Effects model allows both variables to be ordered, but

not necessarily with linear scores.

Fitting models for ordinal variables

Create numeric variables for category scores

PROC GENMOD: Use as quantitative variables in MODEL statement, but not

listed as CLASS variables



Example: Mental impairment and parents’ SES

Srole et al. (1978) Data on mental health status of ∼1600 young NYC residents in

relation to parents’ SES.

Mental health: Well, mild symptoms, moderate symptoms, Impaired

SES: 1 (High) – 6 (Low)

Mental Parents’ SES

health High 2 3 4 5 Low

1: Well 64 57 57 72 36 21

2: Mild 94 94 105 141 97 71

3: Moderate 58 54 65 77 54 54

4: Impaired 46 40 60 94 78 71



2.6 2.0 0.6 -2.4 -4.0

0.7 -1.2

-1.0 -0.6 1.2

-2.6 -3.0 -1.1 0.5 2.4 3.3

Wel

l

Mild

Mod

erat

eIm

paire

dM

enta

l

High 2 3 4 5 Low SES

Mental Impairment and SES: Independence

1.0 -0.6 -1.6

-1.5 1.0

1.3 0.7 0.5 -1.3 -1.3

0.8 -1.1 0.5

Wel

l

Mild

Mod

erat

eIm

paire

dM

enta

l

High 2 3 4 5 Low SES

Linear x Linear

Visual assessment:

Residuals from the independence model show an opposite-corner pattern. This is

consistent with both:

Linear × linear model: equi-spaced scores for both Mental and SES

Row effects model: equi-spaced scores for SES, ordered scores for Mental



Statistical assesment:

Table 8: Mental health data: Goodness-of-fit statistics for ordinal loglinear models

Model G2 df Pr(> G2) AIC AIC-best

Independence 47.4178 15 0.00003 65.4178 35.5227

Col effects (SES) 6.8293 10 0.74145 34.8293 4.9342

Row effects (mental) 6.2808 12 0.90127 30.2808 0.3856

Lin x Lin 9.8951 14 0.76981 29.8951 0.0000

Both the Row Effects and Linear × linear models are significantly better than the

Independence model

AIC indicates a slight preference for the Linear × linear model

In the Linear × linear model, the estimate of the coefficient of aibj is

γ = 0.0907 = log θ, so θ = exp(0.0907) = 1.095.

⇒ each step down the SES scale increases the odds of being classified one step

poorer in mental health by 9.5%.



Fitting these models with PROC GENMOD:mentgen2.sas

1 %include catdata(mental);2 data mental;3 set mental;4 m_lin = mental; *-- copy m_lin and s_lin for;5 s_lin = ses; *-- use non-CLASS variables;6

7 title ’Independence model’;8 proc genmod data=mental;9 class mental ses;

10 model count = mental ses / dist=poisson obstats residuals;11 format mental mental. ses ses.;12 ods output obstats=obstats;13 %mosaic(data=obstats, vorder=Mental SES, resid=stresdev,14 title=Mental Impairment and SES: Independence, split=H V);

Row Effects model:mentgen2.sas

16 proc genmod data=mental;17 class mental ses;18 model count = mental ses mental*s_lin / dist=poisson obstats;19 ...

Linear × linear model:mentgen2.sas

21 proc genmod data=mental;22 class mental ses;23 model count = mental ses m_lin*s_lin / dist=poisson obstats;


Loglinear Models for Categorical Data square

Square-ish tables

Tables where two (or more) variables have the same category levels:

Employment categories of related persons (mobility tables)Multiple measurements over time (panel studies; longitudinal data)Repeated measures on the same individuals under different conditionsRelated/repeated measures are rarely independent, but may have simplerforms than general association

E.g., vision data: Left and right eye acuity grade for 7477 women

Hig

h

2

3

Low

Rig

htE

ye

High 2 3 Low LeftEye

Independence, G2(9)=6671.5



Quasi-Independence

Related/repeated measures are rarely independent— most observations often

fall on diagonal cells.

Quasi-independence ignores diagonals, tests independence in remaining cells

(λij = 0 for i 6= j).

The model dedicates one parameter (δi) to each diagonal cell, fitting them

exactly,


j + δi I(i = j)

where I(·) is the indicator function.

This model may be fit as a GLM by including indicator variables for each

diagonal cell.

DIAG 4 rows 4 cols

1 0 0 00 2 0 00 0 3 00 0 0 4



Using PROC GENMOD· · · mosaic10g.sas

1 title ’Quasi-independence model (women)’;2 proc genmod data=women;3 class RightEye LeftEye diag;4 model Count = LeftEye RightEye diag /5 dist=poisson link=log obstats residuals;6 ods output obstats=obstats;7 %mosaic(data=obstats, vorder=RightEye LeftEye, ...);

Mosaic:

Hig

h

2

3

Low

Rig

htE

ye


Quasi-Independence, G2(5)=199.1



Symmetry

Tests whether the table is symmetric around the diagonal, i.e., mij = mji

As a loglinear model, symmetry is


j + λABij ,

subject to the conditions λAi = λB

j and λABij = λAB

ji .

This model may be fit as a GLM by including indicator variables with equal

values for symmetric cells, and indicators for the diagonal cells (fit exactly)

SYMMETRY 4 rows 4 cols)

1 12 13 1412 2 23 2413 23 3 3414 24 34 4



Using PROC GENMOD· · · mosaic10g.sas

1 proc genmod data=women;2 class symmetry;3 model Count = symmetry /4 dist=poisson link=log obstats residuals;5 ods output obstats=obstats;6 %mosaic(data=obstats, vorder=RightEye LeftEye, ...);

Mosaic:

Hig

h

2

3

Low

Rig

htE

ye


Symmetry, G2(6)=19.25



Quasi-Symmetry

Symmetry is often too restrictive: ⇒ equal marginal frequencies (λAi = λB

i )

PROC GENMOD: Use the usual marginal effect parameters + symmetry:· · · mosaic10g.sas

1 proc genmod data=women;2 class LeftEye RightEye symmetry;3 model Count = LeftEye RightEye symmetry /4 dist=poisson link=log obstats residuals;5 ods output obstats=obstats;

Hig

h

2

3

Low

Rig

htE

ye


Quasi-Symmetry, G2(3)=7.27



Table 9: Summary of models fit to vision data

Model G2 df Pr(> G2) AIC AIC - min(AIC)

Independence 6671.51 9 0.00000 6685.51 6656.23

Linear*Linear 1818.87 8 0.00000 1834.87 1805.59

Row+Column Effects 1710.30 4 0.00000 1734.30 1705.02

Quasi-Independence 199.11 5 0.00000 221.11 191.83

Symmetry 19.25 6 0.00376 39.25 9.97

Quasi-Symmetry 7.27 3 0.06375 33.27 3.99

Ordinal Quasi-Symmetry 7.28 5 0.20061 29.28 0.00

Only the quasi-symmetry models provide an acceptable fit

The ordinal quasi-symmetry model is most parsimonious


Loglinear Models for Categorical Data loglin

ReferencesBickel, P. J., Hammel, J. W., and O’Connell, J. W. Sex bias in graduate admissions: Data from

Berkeley. Science, 187:398–403, 1975.

Friendly, M. SAS System for Statistical Graphics. SAS Institute, Cary, NC, 1st edition, 1991.

Friendly, M. Mosaic displays for multi-way contingency tables. Journal of the AmericanStatistical Association, 89:190–200, 1994.

Friendly, M. Conceptual and visual models for categorical data. The American Statistician, 49:153–160, 1995.

Friendly, M. Extending mosaic displays: Marginal, conditional, and partial views of categoricaldata. Journal of Computational and Graphical Statistics, 8(3):373–395, 1999.

Friendly, M. Visualizing Categorical Data. SAS Institute, Cary, NC, 2000.

Friendly, M. Corrgrams: Exploratory displays for correlation matrices. The American Statistician,56(4):316–324, 2002.

Friendly, M. and Kwan, E. Effect ordering for data displays. Computational Statistics and DataAnalysis, 43(4):509–539, 2003.

Hartigan, J. A. and Kleiner, B. Mosaics for contingency tables. In Eddy, W. F., editor, ComputerScience and Statistics: Proceedings of the 13th Symposium on the Interface, pp. 268–273.Springer-Verlag, New York, NY, 1981.

Srole, L., Langner, T. S., Michael, S. T., Kirkpatrick, P., Opler, M. K., and Rennie, T. A. C. MentalHealth in the Metropolis: The Midtown Manhattan Study. NYU Press, New York, 1978.


Loglinear Models for Categorical Data loglin

Tufte, E. R. The Visual Display of Quantitative Information. Graphics Press, Cheshire, CT, 1983.

van der Heijden, P. G. M. and de Leeuw, J. Correspondence analysis used complementary tologlinear analysis. Psychometrika, 50:429–447, 1985.


Loglinear Models for Categorical Data - YorkU Math … Models for Categorical Data outline Lecture...

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